Recent time-of-flight measurements on muon neutrinos in the OPERA neutrino oscillation experiment have found anomalously short times compared to the light travel-times, corresponding to a superluminal velocity,v − 1 = 2.37 ± 0.32 × 10−5 in units where c = 1. We show that cosmological bounds rule out an explanation involving a Lorentz invariant tachyonic neutrino. At the OPERA energy scale, nucleosynthesis constraints imply v − 1 < 0.86 × 10−12 and the Cosmic Microwave Background observations imply v − 1 < 7.1 × 10−23. The CMB limit on the velocity of a tachyon with an energy of 10 MeV is stronger than the SN 1987A limit. Superluminal neutrinos that could conceivably be observed at particle accelerator energy scales would have to be associated with Lorentz symmetry violation.